One of the main parameters that incoherent scatter radars (ISRs) measure is ion drift velocity v. This can be related primarily to electric field E and neutral wind u, making it possible to use ISR measurements for estimating these parameters, which are of interest when, for example, studying the electrodynamics of the aurorae borealis e.g., determining how auroral currents close within the ionosphere, and studying how energy is transferred between the ionosphere and the neutral atmosphere e.g..

The method for simultaneously estimating E and u for the auroral ionosphere using an incoherent scatter radar was first described by . Since then, this technique has been used and improved seeand references therein. The velocity of both the ion and neutral wind is related to other ionospheric parameters, such as ion-neutral collision frequency and electric field through the momentum equation of the ions. Some parameters can be measured, while others need to be taken from models. The terms with pressure gradients and gravitation are commonly neglected. The electric field can be deduced from measurements higher up where ion-neutral collisions are negligible and used further down by assuming that electric field along the magnetic field line is constant .

Currently ion drifts are measured with a monostatic radar by pointing the transmit beam in three or more different directions and measuring the projection of the ion-velocity vector onto these directions. If the ion velocity then is assumed constant or slowly changing in the horizontal direction for all of the pointing directions, an estimate of the ion drift can be made . If the observations are made with slowly moving dish-based radars, making the measurements in different pointing directions might take several minutes . With modern phased array systems, the scanning time can be reduced .

Multistatic radars can contribute to the measurement challenge by simultaneously measuring a common scattering volume from multiple different directions. Provided there are at least three linearly independent ion velocity measurements, the full vector can be determined. With dish antennas, the ion velocity is obtained only from one intersection volume at a time . In order to obtain measurements along a line or over a volume of space, the antennas need to be steered, which again takes time. In 2008, only the EISCAT UHF system could offer multistatic measurements . Later, the receiver antennas in Kiruna and Sodankylä were converted to receive the signals from EISCAT VHF instead . Earlier, there were also other ISRs that were multistatic .

One of the capabilities of the upcoming EISCAT3D (E3D) incoherent scatter radar is that it can simultaneously observe scatter from ionospheric plasma from at least three different geographically separated receiver sites, each using multiple simultaneous receiver beams that intersect the transmit beam at multiple heights; see Fig. . This is made possible by the use of phased array technology e.g., which allows for fast beam steering and allows the receivers to form multiple simultaneous beams. A measurement of the ion velocity vector along the radar transmit beam is then possible without any other assumptions than homogeneity of ion velocity within the common scattering volumes where the transmit and receive beams intersect .

In this article, we study the ion velocity measurement capability of EISCAT3D. Since ion velocity depends on both electric field, through the Lorentz force, and neutral wind, through collisions, the ion velocity contains information on both electric field and neutral wind. At each range along the transmitted beam, we therefore have six unknowns, three electric field and three neutral wind components, but only three observed ion velocity components. This makes the problem slightly underdetermined. The traditional approach to handle this problem has been to assume a constant electric field along the magnetic field line. Here, we relax this assumption and investigate the physical nature of the problem, showing that physics-based constraints give us additional equations similar to first-order Tikhonov regularization. The ion velocity estimations and the corresponding uncertainty calculations are described in Sect. . The inverse problem of determining electric field and neutral wind from the ion velocities is described in Sect. . In Sect. , model calculations are used to illustrate the resolution and accuracy that are possible.

ISR measurements mainly provide four ionospheric parameters: electron density ne, electron temperature Te, ion temperature Ti, and the ion velocity component along the Bragg scattering vector wp. The connection between wp and the Bragg scattering vector kp for the transmit–receive pair p and the ion velocity v is 1wp=kp⋅v/|kp|+εp, where εp is a random variable that models the velocity measurement errors.

The basis for finding the velocity of the uncharged wind and electric field is through the ion velocity. The measurements wp and the unknown velocity v can be set up as a linear inverse problem : 2w=Kv+ε, where w⊤=[w1,…wP] is a vector containing independent measurements of ion-line Doppler shift, K⊤=[k1…kP] is the theory matrix, and ε is the noise vector. We assume that the noise is independent and identically normal distributed with zero mean and variance of σw2, which means that we assume that the line-of-sight (LOS) ion velocity measured with different receivers has the same uncertainty.

With the first stage of E3D, there will be P=3 measurements of the ion velocity at every range, as shown in Fig. . Since the theory matrix then is quadratic, it will be possible to find v with low uncertainty and without restrictions if the measurements are sufficiently independent linearly see. The Bragg scattering vectors can be calculated from the preliminary positions of E3D as mentioned by . We assume a target in the direction of the magnetic field at ionospheric range, extending outwards from the Skibotn transmitter site.

As it later will become an advantage to have the ion velocity in magnetic field coordinates, we have to transform the scattering vector matrix K. The transformation matrix from geographic to local magnetic coordinates is 3Rgeo→gmag=cos⁡δ-sin⁡δ0sin⁡Isin⁡δcos⁡δsin⁡Icos⁡I-cos⁡Isin⁡δ-cos⁡Icos⁡δsin⁡I, where δ is the declination, and I is the dip angle of the magnetic field . We note that the transform is a rotation, meaning that it is orthonormal, and Rgeo→gmagRgeo→gmag⊤ is equal to the identity matrix. The LOS velocities in geographic coordinates are then related to the ion velocity in local magnetic coordinates as follows: 4w=KRgeo→gmag⊤v+ε.

We find the ion velocity by solving the inverse problem by using the linear least-squares method: 5v^=Rgeo→gmagK⊤K-1K⊤w. The uncertainty of the ion velocity estimate is quantified using the following covariance matrix: 6Σv=Rgeo→gmagK⊤K-1Rgeo→gmag⊤σw2. We will use this uncertainty later when estimating electric field and neutral wind.

The velocity of ion and neutral wind are coupled through collisions as described by the ion momentum equation. This can be found by taking the first moment of the Boltzmann equation e.g. We assume that we can treat the ions as a single fluid. The momentum equation is 15nimidvdt+v⋅∇v=-∇Pi+nimig+qiniE+v×B-∑knimiνikv-vk, where ni is the number density of ions, mi is the ion mass, Pi is the ion pressure tensor, g is the gravitational acceleration, qi is the ion charge, E is the electric field, B is the background magnetic field, νik is the momentum transfer collision frequency between ions and particle species k, and vk is the velocity of particle species k. We assume that spatial variations of the ion velocity are small such that we can neglect the term v⋅∇v. Further we assume that the pressure is isotropic, so we can write the pressure tensor as a scalar pi. Only collisions between ions and neutrals are of importance to change the ion velocity ; other collision terms can be neglected. If the ions obey the ideal gas law, the ion pressure pi can be written as pi=nikBTi. Additionally, we neglect local temperature variations such that ∇Ti=0. Finally, as in previous work, we also neglect the contribution from pressure gradients and gravity. With all these assumptions, Eq. () can be rewritten as 16nimidvdt=qiniE+v×B-nimiνinv-u, where u is the neutral wind velocity.

For steady-state conditions dvdt=0, the ion velocity in the magnetic field coordinate system becomes see 17avx=ux+11+κi2κiBEx-κiuy+κiBEy-κi2ux17bvy=uy+11+κi2κiBEx+κiux+κiBEx-κi2uy17cvz=uz+κiBEz. Here, κi is the ion mobility 18κi=qiBmiνin, where the subscript z denotes the direction along (antiparallel to) the magnetic field, x horizontally towards east, and y perpendicular to the other two directions, giving a right-handed system. Since in this article we are only considering the ion mobility, we will drop the subscript i from now on and just write κ instead.

The component equations can be combined into a compact matrix equation see. 19v=κBCE+Cu, where C is the matrix 20C=11+κ2κ1+κ20-κ1+κ211+κ20001. When estimating the neutral wind and electric field at a certain altitude, Eq. () has to be solved. This is an underdetermined inverse problem with six unknowns, which are all components of both the electric field and the neutral wind velocity. For measurements, we only have the three components of the ion velocity. To resolve this, some a priori assumptions or constraints are required.

The original solution of was to use the fact that κ≫1 at F region altitudes; therefore the ion drift is determined only by the electric field. Then this is assumed to be constant along the magnetic field line. However, the electric field may not be constant in reality e.g.. Such an assumption then affects the neutral wind estimates.

It is possible to assume that the neutral wind and electric field vary smoothly in the whole range of interest and use the full profile of all ion wind measurements to obtain estimates of the neutral wind and electric field. We will outline a procedure to specify a smoothness constraint based on Maxwell's equations in order to give a physically feasible solution.

We start by discretizing the problem as follows: we have a set of ion wind velocity vectors vi…vH which are measurements of Eq. () integrated over a height range defined by the weighting functions di(h) 21vi=∫-∞∞κBCE+Cudi(h)dh+εi, where εi is the noise in measurement i and assumed to be normally distributed with zero mean and covariance described by Eq. () (in magnetic field coordinates). We assume that the unknowns can be described by a set of basis functions 22E(h)=∑j=1NEκ(h)βjbj(h)/B(h) and 23u(h)=∑j=NE+1NE+Nuβjbj(h). This allows Eq. () to be written as 24vi=∑j=1Naij(h)βj+εi, where 25aij=∫-∞∞κ(h)C(h)bj(h)di(h)B(h)dhfor1≤i≤NE∫-∞∞C(h)bj(h)di(h)dhforNE<i≤N, which can be calculated before solving the problem and therefore can be regarded as constants.

We assume that the weighting functions di(h) for the ion wind measurements are boxcars with centre at a certain height and extending exactly halfway to the centre of the nearest box in both directions. At the ends, the measurement height boxes are symmetric around their centre. The basis functions for the unknowns bj(h) are also boxcars. Other basis functions could also be used. We further assume that κ, B, and the rotation matrix C are constant throughout our measurement height boxes d(h).

Equation () in matrix form then can be written as follows: 26V=Ax+ξ, where V⊤=[v1⊤,…,vH⊤], A=a1,1⋯a1,N⋮⋱⋮aH,1⋯aH,N, x⊤=[β1,…,βN], and ξ⊤=[ε1⊤,…,εH⊤].

In order to regularize the problem, we use Gauss' and Faraday's laws. Faraday's law for a static magnetic field, ∇×E=0, gives us three equations for the gradient of the electric field: 27adEydz-dEzdy=027bdExdz-dEzdx=027cdEydx-dExdy=0. Gauss' law for a charge-neutral plasma, ∇⋅E=0, can be written as 28dExdx+dEydy+dEzdz=0.

Equations (a), (b), and () are added to the theory matrix A to regularize the electric field. The derivatives dExdz, dEydz, and dEzdz are approximated with finite differences, with dz equal to the range step.

The horizontal gradients dExdx+dEydy,dEzdx,dEzdy are not specified by our measurements. We therefore treat them as Gaussian random variables ξj,(x,y,z) with zero mean and some variance αj,(x,y,z)-2. Eqs. (a), (b), and () then result in 29Ej,(x,y,z)-Ej+1,(x,y,z)ΔhE=ξj,(x,y,z). These equations are added to the theory matrix. This implies that we assume these three derivatives of the electric field to be smaller than 2/αj,(x,y,z) 95 % of the time. For a box size of ΔhE, this means that (Ej,(x,y,z)-Ej+1,(x,y,z))∼N(0,ΔhE2αj,(x,y,z)-2), which is similar to first-order Tikhonov regularization but with a regularization constant αj,(x,y,z) that varies with both height and electric field component (see, for example, ). It is worth pointing out that the constraints are obtained from Maxwell's equations and therefore have a physical interpretation.

Constraints, such as Eq. (), will favour smoother solutions that are closer to being constant-valued . Throughout this paper, we will loosely use “flatness” to describe how close a function is to a constant value, as the magnitude of the left-hand side of Eq. () is minimized when the function is constant.

For the neutral wind, we also use first-order Tikhonov regularization as described for the electric field above. In addition, we use zeroth-order Tikhonov regularization to restrict the neutral wind to smaller magnitudes. This corresponds to the following statistical assumptions 30uj,(x,y,z)-uj+1,(x,y,z)=ζ1,j,(x,y,z)uj,(x,y,z)=ζ0,j,(x,y,z), where ζ1,j,(x,y,z)∼N(0,Δhu2γ1,j,(x,y,z)-2) and ζ0,j,(x,y,z)∼N(0,γ0,j,(x,y,z)-2). The first row regularizes to the flatness of the profile, and the second constrains the magnitude.

This procedure can be interpreted as adding equations for the derivatives of the unknowns to the theory matrix, where these equal to zero with some uncertainty variance justified by physics. This gives us a problem with smooth, well-behaved solutions provided that the constraints are strong enough.

The regularized linear least-squares solution of the inverse problem is then 31x^=AR⊤Σm-1AR-1AR⊤Σm-1m, where m is the extended measurement vector m⊤=[V⊤0⊤], and AR is the theory matrix A extended with the constraints () and (). We will discuss the measurement error covariance matrix Σm in the next subsection.

In order to demonstrate what a electric field and neutral wind estimate profile could look like, we have simulated a E3D measurement and analysed it. We based the simulations on the Joule II rocket measurement presented in . During the downleg flight, the rocket measured neutral wind at altitudes 90–130 km by tracing chemical releases. The electric field was measured already from 210 km altitude. Since did not include field-aligned components, we used a synthetic profile. We used the electric field and neutral wind profiles to simulate E3D ion velocity measurement with noise added from Eq. (). These simulated ion velocity measurements were used to estimate electric field and neutral wind. By comparing these with the original data set, we can visualize how good the E3D estimates are. The results are shown in Fig. . We use the same regularization schemes as for Fig. .

The results confirm that the electric field is estimated well above 125 km, as predicted by the uncertainty estimates in Sect. . Below 125 km the electric field is not estimated well. For all regularization schemes, the behaviour at lower altitudes is similar. The electric field is estimated to be a constant value corresponding approximately to the value at 125 km altitude, indicating that the regularization contributes with all information of the electric field where ions are demagnetized.

The neutral wind is in general better estimated below 120 km altitude, where it is the largest influencer of the ion wind. Above approximately 125 km, the neutral wind is not well measured in any of the cases. This is not surprising as the neutral wind has little effect on the ion velocity at higher altitudes.

When using the strongest flatness constraints on the electric field, this causes the estimates of the neutral wind to be more fluctuating than the original values (see, for example, Fig. b). We believe that the reason is that the model tries to fit the unknowns to the ion wind measurements but “knows” a priori that the electric field is constant, so all the variation in ion velocity must be explained by the neutral wind instead of the electric field. If the constraints on the electric field are relaxed, the estimate of all unknowns is closer to the original values.

At the heights where the regularization plays a smaller role, the deviation from the original values seems similar to the predicted uncertainties shown in Fig. .

Earlier ISR studies on neutral wind have assumed that the electric field is exactly constant along the magnetic field – mainly due to the lack of three-dimensional ion vector velocity measurements along the whole radar transmit beam. The technique presented in this paper allows us to relax this assumption with a scheme that arises from Maxwell's equations and assumption of horizontal smoothness of electric field. A special case of our regularization scheme is the case where the electric field is approximately constant as a function of height along a magnetic field line. This corresponds to a very strong smoothness assumption on horizontal gradients of electric field (see dark blue line in Figs.  and ). The technique presented in this study can be thus seen as a generalization of the commonly used technique for estimating electric field and neutral wind.

In addition to studies of electric fields and neutral winds separately, a use case of the technique presented here is investigation of Joule heating. For Joule heating, both electric fields and neutral winds are necessary e.g.and references therein. With improved estimates of electric field and neutral wind, the Joule heating can be calculated with higher accuracy.

Our results indicate that it will be possible to observe an altitude profile of electric field and neutral velocity using E3D. However, it is only possible to reconstruct either the electric field or the neutral wind at any given altitude region. This is ultimately due to the fact that above an altitude of approximately 125 km, the ion drift is to a large extent determined by electric field and nearly unaffected by neutral velocity. Similarly, below 125 km, the ion drift is primarily determined by neutral wind.

For future measurements, one important question to solve is what regularization parameters should be used. If the constraints are too weak, the problem is underdetermined and the solution noisy. The classical approach is to assume that the electric field is constant along the magnetic field line. However, the electric field may not always be the constant. Then, as the example shows, the neutral wind estimates have to compensate for variations in the ion drift measurements due to under-resolved electric field variability. Relaxing the assumption of a constant electric field will in these cases improve the results.

Adjusting the regularization constants must be done with caution, since the problem easily becomes underdetermined. Therefore it is important to justify the choices of regularization. For the electric field, we used regularization parameters, which are estimated from in situ rocket measurements. However, the optimal values of the regularization parameters for general use are still to be found. For the values we used, only constraining the electric field was not enough, and we also constrained the gradient of the neutral wind in the same way. Here too, the exact values can be discussed. Forcing the neutral wind velocity gradients to be too small causes the estimates of the neutral wind to fit worse to the ion wind at collision-rich heights. This then increases the noise in the electric field here. If the variation is allowed to be too large, the problem is not solvable. In order to allow for higher variations in the neutral wind but also to use all information we have about it, we added a size constraint of 200 m/s. As can be seen in the uncertainty plots, this restricts the size of the neutral wind components to become smaller.

In future work, the model can, for example, be improved in one of the following ways. If somehow measurements of neutral wind or electric field exist, these can be added additionally as constraints. Such measurements could, for example, be the movement of meteor smoke, polar mesospheric summer echoes, or other measurements of events in the ionosphere that imply size or direction of neutral wind or electric field. An independent measurement of mean neutral wind can often be obtained up to about 100 km using meteor radars e.g..

In this work, we used the same resolution in time for both electric field and neutral wind. The large mass in the neutral atmosphere causes the neutral wind to vary more slowly than the electric field. took advantage of this to use different time resolutions for the different parameters. In the future, it would be an advantage to include this for our model as well.

The technique discussed in this study can be extended further. With the help of phased array technology, E3D will allow fast beam scanning to be used to measure how ion vector velocity and electron density vary within a volume of space. This type of measurement may potentially result in improved estimates of electric field and neutral wind, as more physics-based regularization can be added. We can use Gauss' and Faraday's laws without the need to treat the horizontal gradients as unknown random variables, as they will be determined by the measurements. We can also introduce constraints that are not possible for a one-dimensional profile. It will be possible to apply Ampere's law to enforce current continuity. We can also apply the Navier–Stokes equations to enforce that the neutral wind is approximately consistent with anelastic flow. Estimating electric field and neutral wind within a volume is a topic of future work.